Resources (Updated 2/1)
Dr. Freeman's Daily Schedule - (updated 3/15)
Stewart Textbook Website (Calculus III)
Stewart Textbook Website (Calculus I & II)
Paul's Online Math Notes - Calculus I
Paul's Online Math Notes - Calculus II
Paul's Online Math Notes - Calculus III
Paul's Online Math Notes - Differential Equations
Announcements (Updated 6/8)
Part 2 of the Final Exam can be downloaded here.
The Final Exam will consist of two parts: Part 1 (In-Class Problems) and Part 2 (Take-Home Problems).
Part 1 of the Final Exam will be Wednesday 6/8 from 12:40 PM to 2:25 PM (Periods 5 & 6) in Room 653.
Part 2 of the Final Exam will be available for download by 5 PM on Wed 6/8 and is due by 9 AM on Mon 6/13 (you may submit it electronically).
The "Blank Copies of Old Exams" folder above now contains Exams from this semester.
Exam 3 Solutions are posted below.
Exam 3 will be on Wednesday 5/25, and will (tentatively) begin at 8:35 AM. If you are unable to arrive by 8:30 AM, please let me know no later than in class on Mon 5/23.
Quiz 11 Solutions
Quiz 11 on Friday 5/13 will cover HW 28.5, 29, & 30.
Quiz 10 on Friday 5/6 will cover HW 26 and HW 27.
Exam 2 Solutions are posted below.
Exam 2 will be on Thursday 4/21, and will begin at 8:35 AM.
Quiz 9 (take-home) can be downloaded here and is due Monday 4/18 at 9:00 AM. (Also note the HW due below.)
We will meet at 8:30 AM on Thursday 4/7 instead of Wednesday 4/6 - this week only.
Quiz 7 on Friday 4/1 will cover HW 16 through HW 19.
Quiz 6 on Thursday 3/24 will consist of FCA Page 23 Problem 11 and one other "arg" problem.
Exam 1 will be on Friday 3/11 and will cover (at least) through HW 10.
Starting Wednesday Feb. 10, we will meet every Wednesday at 8:30 AM in Room 735 to discuss homework.Welcome to Advanced Applied Math.
Homework (Updated 6/2)
(You are responsible for all Homework posted and/or updated before 5:00 PM the school day before the due date. Due dates are subject to change, so please check them carefully.)
"ODE" denotes the Ordinary Differential Equations textbook
"FCA" denotes the Fundamentals of Complex Analysis textbook
Notes:
(1) Some of these ODE's are in differential form (i.e., the ODE has already been "multiplied" through by dx). To convert back to the more familiar Leibnitz form, just "divide" the ODE through by dx. Also consider regarding x as the dependent variable and "dividing" through by dy, which may result in an ODE which is easier to solve.
(2) Whenever possible, solve for an explicit solution (isolate the dependent variable), even though the answers in the textbook are often left in implicit form. You will generally be required to find explicit solutions on Exams and Quizzes.
(3) In the ODE Textbook (as well as in Wolfram Alpha, and many other places), "log" denotes the natural logarithm that we normally denote by "ln".
HW 37 is the last homework! We will discuss on Mon 6/6 which portions of it you will be responsible for on the Final Exam.
HW 37 (Complex Integration)
FCA Page 159 Problems 1, 7, 8; Page 170 Problems 3, 6, 7, 13; Page 178 Problem 1; Page 201 Problem 10
Due Mon 6/6
HW 36 (Complex Powers and Inverse Trig Functions Part 2)
FCA Page 136 Problems 10, 11, 12, 15
Due Thu 6/2
HW 35 (Complex Powers and Inverse Trig Functions Part 1)
FCA Page 136 Problems 1, 3, 4, 5, 7
Also, familiarize yourself with the "log" function on Wolfram Alpha for complex arguments, which agrees with the principal value function Log z we have been discussing. For example, compute a few values and check them manually, and check that identities hold where you expect them to hold. You may need to use Wolfram Alpha on the Final Exam.
Due Wed 6/1
HW 34 (Complex Logarithms Part 2)
FCA Page 123 Problems 9, 10, and these Problems
Due Mon 5/23
HW 33 (Complex Logarithms Part 1)
FCA Page 123 Problems 1, 3, 4, 5, 6
Due Fri 5/20
Here is the Solution to HW 23 Problem 14.
HW 32 (Exp, Trig, & Hyp Functions)
FCA Page 115 Problems 1, 11, 12, 13, 15, 17, 20
Due Wed 5/18
HW 31 (Harmonic Functions)
FCA Page 84 Problems 1, 2, 3, 5, 6, 14
Due Mon 5/16
HW 30 (Cauchy-Riemann Equations)
FCA Page 77 Problems 1, 3, 5, 9, 10
Due Fri 5/13
HW 29 (Analyticity)
FCA Page 70 Problems 4, 7, 9, 11, 13
Due Thu 5/12
HW 28.5 (Admissibility)
Do these problems.
Due Mon 5/9
HW 28 (Limits and Continuity)
FCA Page 63 Problems 7, 11, 12, 13, 17, 21, 25
Due Fri 5/6
HW 27 (Functions of a Complex Variable)
FCA Page 56 Problems 1-5, 7c, 8c, 9, 10, 11
Also do Spring 2015 Exam 3 Problem 7
Due Wed 5/4
HW 26 (Planar Sets)
FCA Page 42 Problems 2-8, 11, 12, 15-20
Due Mon 5/2
HW 25 (Reduction of Order)
ODE Page 246 Problems 3, 5, 9, 15
Due Mon 4/18
HW 24 (Series Solutions Near an Ordinary Point)
ODE Page 546 Problems 3, 5, 7, 9, 11
Due Fri 4/15
HW PSR (Power Series Review)
Review the sections on "Power Series" & "Power Series and Functions" in Paul's Online Math Notes - Calculus II and write down any questions you may have.
Due Wed 4/13
HW 23 (Method of Variation of Parameters)
ODE Page 240 Problems 2, 3, 4, 5, 12, 15, 18, 20
Spring 2015 Exam 2 Problem 3 (This is the problem to which I referred at the end of class on Friday.)
Due Mon 4/11
HW 22B (Complex Exponentials are Your Friends)
Do these Problems.
Due Fri 4/8
HW 22A (Method of Undetermined Coefficients)
ODE Page 231 Problems 3, 6, 7, 9, 11, 13, 15, 17, 19, 23, 30, 32
Due Thu 4/7
HW 21 (Linear Homogeneous ODE's with Constant Coefficients - No Restrictions on Roots of Characteristic Equation)
ODE Page 220 Problems 24, 25, 26, 27, 28, 29, 30, 33, 34, 35
Due Mon 4/4
HW 20 (Powers and Roots)
FCA Page 37 Problems 4, 5, 7, 9, 10, 11, 15, 16
Due Thu 3/31 (But please complete as much as possible for Wed 3/30)
HW 19 (Return to Calculus II)
FCA Page 32 Problem 23 (answers are in the back of the book)
Due Wed 3/30
HW 18 (Return to 9th Grade Geometry)
Do this problem.
Due Mon 3/28
HW 17 (Evaluating Complex Trig & Hyperbolic Functions)
FCA Page 115 Problem 5, Spring 2015 Exam 2 Problems 1 & 2
(Note: exp(z) is simply an alternate notation for ez which avoids the need for superscripts)
Due Mon 3/28
HW 16 (Complex Exponentials)
FCA Page 31 Problems 1, 2, 3, 4, 7, 8, 11
Due Thu 3/24
HW 15 (Arg!)
FCA Page 22 Problems 11, 12, 13
Problem 11 is a very nice problem. You will need to use the fact that if z = x + i y, Arg(z) = tan-1(y/x) when x>0 (and only when x>0 !!!), together with the fact that tan-1 is odd.
Also, do:
1) Problems 1 & 3 on Spring 2014 Exam I. (updated 3/22)
Wed 3/23
HW 14 (arg!)
FCA Page 22
Problem 4 (just mimic the induction argument we used in class to prove the similar identity for conjugate)
Problem 5 (if you choose to do the absolute value last, you will do a lot of unnecessary work)
Problem 6 ("cis(theta)" is a ridiculous abbreviation for "cos(theta) + i sin(theta)")
Problem 7
Due Mon 3/21
HW 13 (Welcome to Complex Analysis II)
FCA Page 13 Problems 7, 9, 11, 13, 16
Due Thu 3/17
HW 12 (More Fun with Finite Fields - Quiz 5)
Do these problems. Also in Z mod 11, find all zeroes of the polynomial x^3 + 6 x^2 + 1, use them to factor the polynomial, and then re-multiply the factors as a check.
Due Fri 3/18 (to be turned in, as Quiz 5)
HW MFO (Mixed First Order ODE's)
ODE Page 104 Problems Any/All. You are not responsible for finding integrating factors and/or substitutions for types of ODE's that we did not cover.
Due Thu 3/10
HW 11 (Fun with Finite Fields)
Consider the set {0, 1, 2, 3, 4} with addition and multiplication defined "modulo 5", which means whenever you obtain a number greater than or equal to 5, divide by 5 and replace the number by the remainder, which will always be in {0, 1, 2, 3, 4}. For example: 2 + 4 = 6 = 1 (modulo 5); (3)(4) = 12 = 2 (modulo 5), etc. This, in fact, defines a Field with only 5 members.
a) Construct a 5 x 5 addition table for this Field.
b) Construct a 5 x 5 multiplication table for this Field.
c) Construct a 5 x 2 table listing the additive inverse and the multiplicative inverse for each member of the Field (for which they are defined).
d) Which numbers in this field have square roots, and what are they?
e) Prove that the set {0, 1, 2, 3} with addition and multiplication defined "modulo 4" is, in fact, NOT a Field. (Hint: Repeat steps a), b), and c). What goes wrong?)
However there is a Finite Field with 4 elements with a different addition and multiplication table. Type "finite field of order 4" into Wolfram Alpha to see it.
Due Thu 3/10
HW 10 (Welcome to Complex Analysis I)
FCA Page 5 Problems 7, 9, 11, 13, 14, 17, 19, 20, 21, 23, 24, 25
Due Wed 3/9
HW 9 (Linear Homogeneous ODE's with Constant Coefficients - Characteristic Equation has only Real Roots, Some Possibly Multiple)
ODE Page 220 Problems 1-21 odd, 31, 32 (If you would like more practice, also do Problems 2-8 even, 12-22 even)
Due Mon 3/7
HW 8 (Intro to Linear ODE's)
Do these two problems.
Due Thu 3/3
HW 7 (Linear Independence)
Read ODE Lesson 63B Pages 774-776
For the below problems, you may either use Wronskians or the definition of Linear Dependence given in class.
1) Prove that any two non-constant functions which differ by a (non-zero) constant are Linearly Independent.
2) Prove that any set of functions that includes the (constant) zero function is Linearly Dependent.
3) Prove that the three functions: cos x, (cos x)^3, and sin x sin 2x are Linearly Dependent. (^3 means cubed)
Due Wed 3/2
HW 6 (Exact First Order ODE's)
ODE Page 79 Problems 5-17 odd. In Problem 7, "dy" should be outside the parentheses.
Due Mon 2/29
HW 5 (Bernoulli Equation)
ODE Page 97 Problems 3, 7, 8, 22 (done below as separable, do again as Bernoulli and compare), 23
Due Thu 2/25
HW 4 (Homogeneous First Order ODE's)
ODE Page 61 Problems 3, 5, 6, 8, 12-15 (solve explicitly for y in 12-15) (Hint: Problem 3 is much easier if x is regarded as the dependent variable and you let u = x/y)
Due Wed 2/24
HW 3B (Linear First Order ODE's - continued)
1) ODE Page 97 Problems 21, 22, 24. Solve explicitly for the solution y = f(x) and determine the interval of validity. (CORRECTED - do not do 23)
2) Do this problem.
Due Mon 2/22
HW 3A (Linear First Order ODE's)
ODE Page 97 Problems 1, 2, 5, 6, 9, 11, 13, 14, 17
Due Fri 2/12
HW 2B (Separable First Order ODE's)
ODE Page 55 Problems 10, 12, 18, 19, 20, 21
Due Thu 2/10
HW 2A (Separable First Order ODE's)
ODE Page 55 Problems 2, 3, 5, 7, 11, 17
Due Wed 2/10
HW 1 (Verifying Solutions to ODE's)
ODE Page 27 Problems 1-3; Page 37 Problems 1-4
Due Fri 2/5
Exam Results (Updated 5/26)
Note: All Test Statistics are before adding make-up points (if applicable).
Exam 1
Statistics
Student Work
Exam 2
Statistics
Solutions (Solution to Problem 3 updated on 4/22)
Student Work
Exam 3
Statistics
Student Work